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Robust Wald Tests in SUR Systems with Adding Up Restrictions: An Algebraic Approach to Proofs of Invariance


  • Surajit Ray

    (University of Iowa)

  • B. Ravikumar

    (University of Iowa)

  • N. Eugene Savin

    (University of Iowa)


In this paper, we examine the robust Wald test statistic for SUR systems with adding up restrictions where the same explanatory variables are present in all equations and where heteroskedasticity and/or autocorrelation of unknown forms may be present. For this case, the coefficients are usually estimated by least squares, equation by equation. For testing the typical hypotheses of interest, we show that the robust Wald statistic, i.e., the statistic based on the heteroskedasticity and autocorrelation consistent covariance matrix estimator, is invariant to the equation deleted. Our proof of invariance is algebraic and does not rely on parametric assumptions or on the knowledge of the covariance matrix of disturbances. Furthermore, the adding-up restrictions we consider are of a general form: the weighted sum of the dependent variables adds up to one of the explanatory variables, not necessarily a constant. We illustrate our results using the Capital Asset Pricing Model.

Suggested Citation

  • Surajit Ray & B. Ravikumar & N. Eugene Savin, 1998. "Robust Wald Tests in SUR Systems with Adding Up Restrictions: An Algebraic Approach to Proofs of Invariance," Econometrics 9802002, EconWPA.
  • Handle: RePEc:wpa:wuwpem:9802002
    Note: Type of Document - Postscript; pages: 22 ; figures: included

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    References listed on IDEAS

    1. Barten, A. P., 1969. "Maximum likelihood estimation of a complete system of demand equations," European Economic Review, Elsevier, vol. 1(1), pages 7-73.
    2. Mandy, David M. & Martins-Filho, Carlos, 1993. "Seemingly unrelated regressions under additive heteroscedasticity : Theory and share equation applications," Journal of Econometrics, Elsevier, vol. 58(3), pages 315-346, August.
    3. Ravi Jagannathan & Ellen R. McGrattan, 1995. "The CAPM debate," Quarterly Review, Federal Reserve Bank of Minneapolis, issue Fall, pages 2-17.
    4. White, Halbert, 1980. "A Heteroskedasticity-Consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity," Econometrica, Econometric Society, vol. 48(4), pages 817-838, May.
    5. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 33(1), pages 125-132.
    6. Berndt, Ernst R & Savin, N Eugene, 1975. "Estimation and Hypothesis Testing in Singular Equation Systems with Autoregressive Disturbances," Econometrica, Econometric Society, vol. 43(5-6), pages 937-957, Sept.-Nov.
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    Cited by:

    1. Javed Iqbal & Robert Brooks & Don Galagedera, 2010. "Multivariate tests of asset pricing: simulation evidence from an emerging market," Applied Financial Economics, Taylor & Francis Journals, vol. 20(5), pages 381-395.
    2. Iqbal, Javed & Brooks, Robert & Galagedera, Don UA, 2007. "Robust Tests of the Lower Partial Moment Asset Pricing Model in Emerging Markets," MPRA Paper 25349, University Library of Munich, Germany, revised May 2007.

    More about this item


    SUR System; Adding up; Wald test; Heteroskedasticity; Autocorrelation;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General


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